Full-back Cam Winnett strikes first for visitors.
But front-rows Andre-Hugo Venter and Ntuthuko Mchunu hit back for hosts seeking revenge for loss at Cardiff in final game of regular season.
Wing Leolin Zas grabs third Stormers try after Keiron Assiratti yellow card before half-time.
Taine Basham scores early in second half and James Botham follows to keep Cardiff in tie.
But poor kick hands Sacha Feinberg-Mngomezulu easy score before driving lineout earns fifth try.
JJ Kotze crosses late to send Stormers into semi-final against either Leinster or Lions - who meet in Dublin this evening (20:00 BST).
Get Involved: Use yellow button on page.
FT: Stormers 44-21 Cardiff
Cardiff wing Josh Adams speaking to S4C:
"We're disappointed with the result, but throughout the season we've been great.
"We deserved to be in a game like this today.
"We battled really hard and right to the end.
"But we're a young team and there's a lot of growth to be done.
"I'm looking forward to next season and hopefully to improving as well."
FT: Stormers 44-21 Cardiff
Former Wales and Cardiff fly-half Rhys Patchell speaking to S4C:
"Going to South Africa is hard, whenever you go, going at this point is tough when those players are at their best.
"In the end it was a fair result."
Image source, Rex FeaturesFT: Stormers 44-21 Cardiff
Former Wales hooker Ken Owens speaking to S4C:
"They found their way back into the game and were able to start attacking.
"I feel they'll look back at the game when they were back in it, and a mistake from Ioan Lloyd, that took the game away from them a bit.
"But they battled until the final minutes. They can be proud."
FT: Stormers 44-21 Cardiff
Gareth Delve
Former Wales international on BBC Radio Wales
I think there'll be a lot for Cardiff to take away from that.
There'll be pride at the effort.
The set-piece dominance was just too much in the end and the lack of discipline and ability to manage that area of the game has cost Cardiff a famous opportunity at a victory.
In open play they caused problems, but this Stormers team just had too much for them on the day.
Image source, Rex FeaturesThat's it.
The Welsh club season ends with Stormers beating Cardiff to earn a place in the URC semi-finals.
They will either travel to Leinster or stay in South Africa to face Lions.
Jurie Matthee
Eoghan Cross can't let this game just drift off. Another penalty and Jurie Matthee knocks over the kick with the clock in the red.
Stormers 41-21 Cardiff
Gareth Delve
Former Wales international on BBC Radio Wales
Some tired bodies in that Cardiff defensive line.
Just too many holes in that line and too much quality in that Stormers team.
They've defended manfully, but it's just been a bridge too far in these last 10 minutes.
JJ Kotze (Con: Jurie Matthee)
Stormers are in the semis.
Cardiff are out on their feet and, after an interception by Ruhan Nel snaffles possession, the hosts go on the attack in the 22.
Replacement hooker JJ Kotze goes through a huge hole in the defence, with Jurie Matthee adding the extras.
Stormers 34-21 Cardiff
Cardiff had a glimmer after Ioan Lloyd chases his own grubber but Stormers get back to cover.
The Blue and Blacks need to create a try from nowhere to make this a nervy finish.
Image source, Rex FeaturesStormers 34-21 Cardiff
Javan Sebastian returns and Cardiff have 15 men for the last seven minutes.
They still have a slim chance but the clock is ticking.
Stormers 34-21 Cardiff
Stormers think that they have sealed the win but the try by Stefan Ungerer is chalked off for obstruction, with Ruhan Nel preventing Ben Thomas from making a tackle at a ruck in the build-up.
Stormers 34-21 Cardiff
It's a strange situation where Cardiff are down by 13 points but are having to kill the clock.
One more score will do it for Stormers.
Stormers 34-21 Cardiff
Gareth Delve
Former Wales international on BBC Radio Wales
It's just an absolute nightmare for Cardiff.
It's a big hill to surmount now, but they need to stick with it.
Image source, Huw Evans Picture AgencyUse yellow button on page
Now they are checking every try scored...why wasn't the first Stormers try checked this carefully? Lack of consistency in URC refereeing is so frustrating!
Nick, Wales
Jurie Matthee
Jurie Matthee makes no mistake from a central position, just outside the 22.
Cardiff's 13 men have to somehow stay in this before getting back to 15 and giving it one last blast in the closing stages.
Feels like the game is gone though.
Stormers 31-21 Cardiff
Another penalty against Cardiff, with Josh McNally trappped on the Stormers side of a ruck.
The hosts will try to stretch their lead from the tee.
Stormers 31-21 Cardiff
Gareth Delve
Former Wales international on BBC Radio Wales
It's a double whammy for Cardiff with that yellow card beforehand.
Stormers 31-21 Cardiff
Nightmare for Cardiff, who are now down to 13 men due to uncontested scrums.
Javan Sebastian is in the sin bin and Keiron Assiratti went off injured.
They've had to take off wing Jacob Beetham and number eight Taine Basham to get enough front rowers on the pitch, even though it's uncontested scrums.
Cardiff have a mountain to climb now.
Paul de Villiers
No stopping Stormers this time.
Line-out won and flanker Paul de Villiers is the man to go over from the powerful drive.
The conversion is wide but Cardiff are 10 points down and only have 14 men.
Stormers 26-21 Cardiff
Cardiff hit the drive early before Stefan Ungerer is held up over the line.
Prop Javan Sebastian becomes the second Cardiff prop to be sin-binned.
Stormers missed a trick there - a scrum would have got the visitors down to 13 men due to uncontested scrums.
